The beta plane approximation accurately models the latitudinal variation of the Coriolis parameter
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Some studies use the beta-plane approximation to model latitudinal variations for specific frameworks, while other research derives leading-order equations using different approaches like thin-shell approximations.
Pressure anomaly set by the open ocean affects the dynamic topography and associated circulation over the continental shelf, which is explored here on a linearized β-plane arrested topographic wave framework that considers the variation in Coriolis parameter with latitude. It was found that on a meridional shelf, a nondimensional parameter Peβ, termed the β Péclet number, signifies the characteristics of open ocean–shelf interaction. The Peβ ≡ Dβ/α is determined by the ratio of long-wave-limit planetary to topographic Rossby wave speeds, i.e., the β drift Dβ, and the linear Ekman number α. On the western boundary shelf, due to the westward planetary Rossby wave, open ocean pressure propagates shoreward as Peβ > 1, and shelf circulation peaks where Peβ drops to 1. At this location, the planetary β effect is balanced by the bottom friction. The Peβ = 1 must occur either on the shelf or on the coastal wall when Peβ > 1 is observed at the shelf edge. On the eastern boundary shelf, however, Peβ < 0, the pressure anomaly is removed from the shelf, and hence the inductive circulation decays rapidly from the shelf edge. This β effect is robust on gently sloping meridional shelves. For zonal shelves, the planetary β increases the effective bottom slope on the northern boundary shelf but decreases it on the southern one, in a sense of potential vorticity conservation. However, this effect could be less significant in reality, given the complex dynamics involved. The above mechanism can explain the dynamics driving the Taiwan Warm Current in the East China Sea and its bifurcation around 28°N.
The leading-order equations governing the unsteady dynamics of large-scale atmospheric motions are derived, <i>via</i> a systematic asymptotic approach based on the thin-shell approximation applied to the ellipsoidal model of the Earth's geoid. We present some solutions of this single set of equations that capture properties of specific atmospheric flows, using field data to choose models for the heat sources that drive the motion. In particular, we describe standing-waves solutions, waves propagating towards the Equator, equatorially trapped waves and we discuss the African Easterly Jet/Waves. This work aims to show the benefits of a systematic analysis based on the governing equations of fluid dynamics.
The leading-order equations governing the unsteady dynamics of large-scale atmospheric motions are derived, via a systematic asymptotic approach based on the thin-shell approximation applied to the ellipsoidal model of the Earth’s geoid. We present some solutions of this single set of equations that capture properties of specific atmospheric flows, using field data to choose models for the heat sources that drive the motion. In particular, we describe standing-waves solutions, waves propagating towards the Equator, equatorially trapped waves and we discuss the African Easterly Jet/Waves.
Retaining the full nature of the Earth’s curved-space geometry is essential for large-scale atmospheric flows. To go beyond the limitations of the flat geometry of the f -plane approximation, the typical approach consists in invoking a weak contribution from the curvature by using the β -plane approximation (see [ 1 ]). However, in contrast to the f -plane approximation, the β -plane approximation fails to represent a consistent approximation to the governing equations for geophysical flows at mid-latitudes and in polar regions; see [ 2 ].
A further typical simplification in research investigations is to ignore the fine structure of the density variation by introducing weighted averages or by relying on the Boussinesq approximation. As demonstrated by [ 3 ] in the case of steady flow, a detailed and extensive description of the motion in an atmosphere that envelopes an ellipsoidal model of the geoid, superimposed on a stationary but variable background state, can be formulated and solved. We now show that the same approach can be adopted when time dependence is added to the system, which, in this initial phase of the development of a coherent mathematical description, is based on an idealized model of the atmosphere.
This fact is a strong motivation to derive a leading-order generic system of (reduced) equations, based on reliable and transparent approximation procedures, as pursued in the present paper. Unsurprisingly, we recover qualitative features that are similar to those encountered in classical models, but there are a few important corrections to be noted, especially with regards to the dispersion relation of zonally harmonic waves and concerning the meridional decay rate of equatorially trapped waves (see §5).
This situation is somewhat similar to that encountered in the investigation of Rossby waves: the beta-plane approximation was the step forward from the f -plane approximation that revealed to Rossby [ 7 ] the presence of these waves, but Haurwitz [ 8 ] extended the accuracy (especially with regard to the meridional extent of the wave) by taking the spherical shape of the Earth into account (see also [ 9 , 10 ]). 2 .
(b) Equatorially trapped waves Due to the nature of their energy sources, as well as the small Coriolis forcing, large-scale equatorial atmospheric flows present specific structural features. Synoptic-scale disturbances of the atmosphere outside the tropics are mainly driven by horizontal temperature gradients, the primary energy source being the potential energy associated with the latitudinal temperature gradient [ 19 ]. In the tropics, however, horizontal temperature gradients are very small, and the energy harnessed by atmospheric flows comes from diabatic heating due to latent heat release, mostly occurring in association with moist convection.
The classical theoretical approach for describing equatorially trapped waves (see [ 19 ]) concentrates on their horizontal structure, using a shallow water model (for a fluid system of mean depth h e in a motionless basic state) and the equatorial β -plane approximation. While the predictions made by relying on these approximations (non-dispersive, eastward propagating trapped waves) are somewhat similar to ours, there are significant differences. Firstly, approximating sin θ and cos θ by the first two terms in their Taylor expansions near θ = 0 leads to an exponential meridional decay rate; ours is not so severe because we have not approximated the θ -dependence.
Thus this aspect of the problem is included in the general formulation, but ignored in the main thesis that we propound here.) As we found in our earlier work [ 3 ], the real surprise is that the thin-shell approximation, without recourse to any other assumptions about the nature of the flow, leads to a complete description of the dominant dynamic and thermodynamic elements that are needed to describe the atmosphere. This earlier work showed how standard steady-state models that are used for specific problems in the atmosphere (e.g.
This conceptually coherent modelling brings to light the significant dependency of the atmospheric flow on the heat forcing, avoiding adjustments of model parameterizations that often mask underlying deficiencies of models, which are not derived systematically from the governing equations, a procedure that may fail to capture the relevant physical processes even if, due to parameter calibration, they might exhibit agreement between simulation and data. Our equations admit solutions that are harmonic in time, and so the majority of our examples are based on solutions that possess this property.
Knowing the composition of Jupiter's atmosphere is crucial for constraining Jupiter's bulk metallicity and formation history. Yet, constraining Jupiter's atmospheric water abundance is challenging due to its potential nonuniform distribution. Here, we explicitly resolve the water hydrological cycle in Jupiter's midlatitudes using high-resolution simulations. Falling precipitation leads to a significant large-scale depletion of water vapor beneath the lifting condensation level. A nonuniform water vapor distribution emerges in the midlatitude simulation with a changing Coriolis parameter across latitudes and spatially uniform cooling and heating. Water abundance at the 7-bar level varies by up to a factor of ten across latitudes, from subsolar to supersolar values. We propose that nonlinear large-scale eddies and waves tend to drift air parcels across latitudes along constant potential vorticity (PV) surfaces, thereby sustaining latitudinal dependencies in water vapor and the interplay between water distribution and large-scale dynamics. Therefore, water distribution is influenced by the vertical structure of density stratification and changing Coriolis parameter across Jupiter's midlatitudes, as quantified by PV. Additionally, the water hydrological cycle amplifies the specific energy of air parcels through the latent heat effect, thereby slowing down vertical mixing with a latent heat flux. The horizontal gradient of water is expected to be more pronounced with a supersolar water abundance. We suggest that similar interplays between precipitating condensates, planetary rotation, and distribution of condensable species generally exist in the weather layer of fast-rotating giant planets. The ongoing Juno mission and future Uranus mission may further reveal the nonuniform distribution of condensed species and their interplay with large-scale dynamics.
A nonuniform water vapor distribution emerges in the midlatitude simulation with a changing Coriolis parameter across latitudes and spatially uniform cooling and heating. Water abundance at the 7-bar level varies by up to a factor of ten across latitudes, from subsolar to supersolar values. We propose that nonlinear large-scale eddies and waves tend to drift air parcels across latitudes along constant potential vorticity (PV) surfaces, thereby sustaining latitudinal dependencies in water vapor and the interplay between water distribution and large-scale dynamics.
III) How can we improve our understanding of chemical distributions in planetary atmospheres, especially those that matter to measuring metallicity? We will first present our findings from the numerical simulation and then provide explanations to address the listed questions. The Simulated Water Distribution We simulated the water vapor distribution in Jupiter’s midlatitudes using the nonhydrostatic model SNAP ( 26 , 38 , 39 ) in a Cartesian box with linearly varying Coriolis parameter across the latitude (called beta-plane approximation). The simulation domain spans 45,000 km in latitude (y-direction) and 60,000 km in longitude (x-direction).
We convert the distances in x − and y − directions to equivalent longitude and latitude with the radius of Jupiter and local Coriolis parameter on the β plane (see Fig. 1 caption). Vertically, the domain extends from ∼ 87 bar ( − 240 km) at the lower boundary to ∼ 0.002 bar (100 km) at the top, with the reference height ( z 0 = 0 km) set at 1 bar. The spatial resolution is 150 km in both longitude and latitude (about 0 . 1 ° ) and 2 km in the vertical direction. The spatial resolution allows us to resolve the Rossby deformation radius and the scale height of water vapor. The initial water vapor abundance in the deep reservoir is assumed to be three times solar and horizontally uniform.
We propose that nonlinear large-scale eddies and waves which can propagate within the stratified atmosphere with a changing Coriolis parameter across latitudes (called the β effect by meteorologists) sustain the nonuniform distribution of water vapor shown in Figs. 1 A and D , 3 A , and 4 A . In our simulations, the latitudinal gradient of water vapor initially forms from a large-scale convective mixing during the spin-up phase ( Movie S2 ). Subsequently,
The definition of PV in the context of Jupiter’s atmosphere could be, [4] PV = ( 2 Ω + ∇ × u ) ⏟ absolute vorticity · ∇ θ v ρ ⏟ weighted stratification ≈ ( f + ζ ) 1 ρ ∂ θ v ∂ z = ( f + ζ ) θ v ρ g N 2 , where Ω is the angular velocity of the planet, f = 2 Ω sin ϕ = f 0 + β 0 y is the Coriolis parameter that increases in latitude ϕ , θ v is the virtual potential temperature that represents the potential density of the air parcel adjusted by isentropic expansion or compression (i.e., buoyancy), ζ = ( ∇ × u ) · z ^ is the vertical component of relative vorticity, g is the gravity. The approximation in Eq.
4 could represent more than 99% of total PV, while the rest only contributes to less than 1% ( SI Appendix , Fig. S6 ). We find the leading terms in Eq. 4 that govern the simulated PV structure in Figs. 3 and 4 are the Coriolis parameter f , which increases in latitude, and the stratification N 2 , which is set by precipitation and peaks near 7.5 bars ( Fig. 4 B ). The variations of the other quantities are generally more minor than the change of f and N 2 across the domain (see Fig. 4 caption for numbers).
The meridional gradient of the Coriolis parameter (i.e., planetary vorticity), β 0 , is fixed as a constant for the β -plane simulation, β 0 = 2 Ω / a ≈ 4.92 × 10 − 12 m − 1 s − 1 , where a is Jupiter’s planetary radii. Hence, the Coriolis parameter, f , is linear in latitude and calculated by f = 2 Ω sin ϕ 0 + β 0 y , where ϕ 0 = 20 ° N is the southern boundary of the domain and y is the meridional distance from the southern boundary. We adopt reflecting and free-slip boundary conditions in the meridional and vertical directions, but periodic boundary conditions in the longitudinal direction.
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